Simplify the following expression and state the condition under which the simplification is valid. $x = \dfrac{-9k^2 - 27k + 162}{4k^2 - 24k + 36}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ x = \dfrac {-9(k^2 + 3k - 18)} {4(k^2 - 6k + 9)} $ $ x = -\dfrac{9}{4} \cdot \dfrac{k^2 + 3k - 18}{k^2 - 6k + 9} $ Next factor the numerator and denominator. $ x = - \dfrac{9}{4} \cdot \dfrac{(k - 3)(k + 6)}{(k - 3)(k - 3)}$ Assuming $k \neq 3$ , we can cancel the $k - 3$ $ x = - \dfrac{9}{4} \cdot \dfrac{k + 6}{k - 3}$ Therefore: $ x = \dfrac{ -9(k + 6)}{ 4(k - 3)}$, $k \neq 3$